Abstract.A number of nonlinear models have recently been proposed for simulating soil carbon decomposition. Their predictions of soil carbon responses to fresh litter input and warming differ significantly from conventional linear models. Using both stability analysis and numerical simulations, we showed that two of those nonlinear models (a two-pool model and a three-pool model) exhibit damped oscillatory responses to small perturbations. Stability analysis showed the frequency of oscillation is proportional to ε −1 − 1 K s /V s in the two-pool model, and to ε −1 − 1 K l /V l in the threepool model, where ε is microbial growth efficiency, K s and K l are the half saturation constants of soil and litter carbon, respectively, and V s and V l are the maximal rates of carbon decomposition per unit of microbial biomass for soil and litter carbon, respectively. For both models, the oscillation has a period of between 5 and 15 years depending on other parameter values, and has smaller amplitude at soil temperatures between 0 and 15 • C. In addition, the equilibrium pool sizes of litter or soil carbon are insensitive to carbon inputs in the nonlinear model, but are proportional to carbon input in the conventional linear model. Under warming, the microbial biomass and litter carbon pools simulated by the nonlinear models can increase or decrease, depending whether ε varies with temperature. In contrast, the conventional linear models always simulate a decrease in both microbial and litter carbon pools with warming. Based on the evidence available, we concluded that the oscillatory behavior and insensitivity of soil carbon to carbon input are notable features in these nonlinear models that are somewhat unrealistic. We recommend that a better model for capturing the soil carbon dynamics over decadal to centennial timescales would combine the sensitivity of the conventional models to carbon influx with the flexible response to warming of the nonlinear model.

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We consider regular (identical-edge identical-node) networks whose cells can be grouped into classes by an equivalence relation. The identification of cells in the same class determines a new network -the quotient network. In terms of the dynamics this corresponds to restricting the coupled cell systems associated with a network to flow-invariant subspaces given by equality of certain cell coordinates. Assuming a bifurcation occurs for a coupled cell system restricted to the quotient network, we ask how that bifurcation lifts to the overall space. Surprisingly, for certain networks, new branches of solutions occur besides the ones that occur in the quotient network. To investigate this phenomenon we develop a systematic method that enumerates all networks with a given quotient. We also prove necessary conditions for the existence of solutions branches not predicted by the quotient. We then apply our method to two particular quotient networks; namely, two-and three-cell bidirectional rings. We show there are no additional bifurcating solution branches when the quotient network is a two-cell bidirectional ring. However, two of the 12 five-cell networks that have the three-cell bidirectional ring as a quotient network * CMUP is supported by FCT through the programmes POCTI and POSI, with Portuguese and European Community structural funds. 1 exhibit bifurcating solutions that do not occur in the quotient itself. Thus, network architecture sometimes forces the existence of bifurcating branches in addition to the ones determined by the quotient.

A cell is a system of differential equations. Coupled cell systems are networks of cells. The architecture of a coupled cell network is a graph indicating which cells are identical and which cells are coupled to which. In this paper we continue the work of Stewart, Golubitsky, Pivato and Török by classifying all homogeneous three-cell networks (where each cell has at most two inputs) and classifying all generic codimension one steady-state and Hopf bifurcations from a synchronous equilibrium. We use combinatorial arguments to show that there are 34 distinct homogeneous three-cell networks as opposed to only three such two-cell networks.We show that codimension one bifurcations in homogeneous three-cell networks can exhibit interesting features that are due to network architecture. Indeed, network architecture determines, even at linear level, the kind of generic transitions from a synchronous equilibrium that can occur as we vary one parameter and plays a crucial role in establishing how the solutions on the bifurcating branches manifest themselves in each cell.Mathematics Subject Classification: 34A34, 34C14, 34C15, 37G15, 34K18

Abstract. A number of nonlinear models have recently been proposed for simulating soil carbon decomposition. Their predictions of soil carbon responses to fresh litter input and warming differ significantly from conventional linear models. Using both stability analysis and numerical simulations, we showed that two of those nonlinear models (a two-pool model and a three-pool model) exhibit damped oscillatory responses to small perturbations. Stability analysis showed the frequency of oscillation is proportional to √ (ϵ −1−1)Ks/Vs in the two-pool model, and to √ (ϵ −1−1)Kl/Vl in the three-pool model, where ϵ is microbial growth efficiency, Ks and Kl are the half saturation constants of soil and litter carbon, respectively, and Vs and Vl are the maximal rates of carbon decomposition per unit of microbial biomass for soil and litter carbon, respectively. For both models, the oscillation has a period between 5 and 15 yr depending on other parameter values, and has smaller amplitude at soil temperatures between 0 °C to 15 °C. In addition, the equilibrium pool sizes of litter or soil carbon are insensitive to carbon inputs in the nonlinear model, but are proportional to carbon input in the conventional linear model. Under warming, the microbial biomass and litter carbon pools simulated by the nonlinear models can increase or decrease, depending whether ϵ varies with temperature. In contrast, the conventional linear models always simulate a decrease in both microbial and litter carbon pools with warming. Based on the evidence available, we concluded that the oscillatory behavior and insensitivity of soil carbon to carbon input in the nonlinear models are unrealistic. We recommend that a better model for capturing the soil carbon dynamics over decadal to centennial timescales would combine the sensitivity of the conventional models to carbon influx with the flexible response to warming of the nonlinear model.